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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"02a4f2156e7418c7b52fcec7c258598de880879654dd0b587a3b3eb791369857","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-849de7332df28657","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        1;\n  1:        1,          1;\n  2:        3,          7,           2;\n  3:       15,         65,          43,           6;\n  4:      105,        775,         905,         286,          24;\n  5:      945,      11375,       21070,       11389,        2126,         120;\n  6:    10395,     199045,      552860,      456701,      142744,       17676,        720;\n  7:   135135,    4053665,    16320535,    19397854,     9049411,     1847656,     163332,      5040;\n  8:  2027025,   94315375,   538365905,   887080586,   580505149,   174016368,   25071164,   1664496,    40320;\n  9: 34459425, 2470442975, 19691645050, 43876838929, 38750498171, 16013561157, 3346199956, 358800156, 18568944, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+n*x)*P(n-1,x) + n*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n - 1 + n*x)*P[n - 1, x] + n*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n - 1 + n*x)*v[i - 1] + n*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-849de7332df28657","steps":["basis A008277"],"edges":["e-2164c90874641e81"],"text":"A088996 \u2192 basis A008277"},"headline":"A088996 \u2192 basis A008277","chain":null}